Abstract
The
subject of tuning and temperament continues to provide a never ending source of
interest and income for a constant stream of academics. Because it
requires just that little extra effort to comprehend the necessary simple
arithmetic (it is wrong to dignify it as mathematics), it is easy for those with the
inclination to wrap up their work in a cloak of mystery and authority which is
actually largely spurious. It disguises the fact that many of the claims
made about the temperaments favoured by Bach, say, are completely unproveable.
In reality, they fall into the same category as the story that he once found some
coins in fish heads thrown out of the window of an inn, and they are about as useless.

This
article shows that an uncomfortable proportion of recent work on temperament is unscholarly in that it
projects today's understanding, values and culture several centuries backwards
as though these things have never changed. Thus the authors of such material are merely wallowing, apparently unconsciously, in a
sea of reverse anachronism. They are literally out of time. Some are
also apparently unconscious of the errors in their work. By looking at the
realities of musical life in the 17th and 18th centuries it is suggested that
some if not many contemporary temperaments can be traced to the fact that
stringed keyboard instruments had wood frames, with the consequential tuning
instability this implies. Also the role possibly played by impure octaves
in these temperaments is examined.

Because
a correspondent drew to my attention an article on temperament of such utter
fatuousness, carried by a journal with learned pretensions which has clearly
abandoned any attempt at peer review, I now rest easy in the knowledge that
anything I might write on the subject could not possibly be any worse.I am grateful to him.

To
study temperament is to read the entire story of music itself across the world
since the dawn of time.Melodic as opposed to percussive musical instruments arose
firstly because of the aural observation of the first three natural harmonics in
the sounds of pipes.These were
then given a separate existence by tuning additional strings or pipes to the
same frequencies.The intervals
between these would have been octaves and fifths because of their genesis from
the natural harmonic series.As
time went on it was found that the fifth alone had the magical property of
generating eleven notes which could be used to fill the gap between two octavely
related ones, thereby giving us the chromatic scale still used today.Simultaneously, it became obvious that the frequencies of these notes had
to be adjusted from what was regarded as their “pure” values so they could
fit into the octave.Thus arose an
appreciation of the root problem of temperament together with the concepts of
scale and key, and the basic architectures of all melodic instruments.In turn, these tools have given us all of music.

Yet
the overlay of history often applied to this simple story is perverse and
skewed.It distorts the beauty of a
narrative which is nothing less than transcendental. Today’s practitioners of
temperament, the Temperamentalists, sometimes make a particularly crude blunder
by translating today’s understanding of the physics of music, together with
other aspects of our culture and values, several centuries backwards as though
they have never changed.This has
facilitated the generation of the quantity of material we have today on subjects
such as “Bach’s Temperaments”.When
assimilating such material it is therefore essential to understand this
background before reaching a true verdict of its worth. This self-indulgent
reverse anachronism on the part of some Temperamentalists is reflected in the
title of this article.

The
article is divided into four parts.Part
1 shows that, although the ancient Greeks are generally credited with much of
today’s understanding of temperament, most of it was in fact well in place by
the time of Pythagoras.Part
2 looks at some of the realities of musical life in Europe in the 17th
and 18th centuries, avoiding the anachronistic translation referred
to above.It relates the genesis of
contemporary temperaments to the single fact that stringed keyboard instruments
then had wood frames, with the consequential tuning instability.Part 3 next
reviews some current work in temperament, demonstrating among other things a
disgraceful shoddiness in much of it which is reflected in the level of
numerical error.Finally, Part 4
considers whether impure octaves should form part of modern studies of early
temperaments rather than being virtually ignored, and if so, how they should be
approached.

There
is a breed of scholar turned out by university departments of the humanities
which has apparently been trained to believe that it is OK for them to ignore
things which do not interest them or about which they know little.Some historians are a case in point.It is incredible that a recent three-volume 1500-page “History of
Britain” could have appeared which virtually overlooks the contribution of our
scientists and engineers to national and world culture [7], just like the TV
series which spawned it.I thought
that history seen only in terms of dates, kings and battlefields is regarded as
juvenile and superficial today, as well as boring and passé.In similar vein is the “Cambridge Biographical Encyclopedia” which
devotes less space to Michael Faraday than to James Curley [8].And who was he?He was a
mayor of Boston (Massachusetts). Hands up those who have heard of him.Both these examples illustrate a perverse view of the relative importance
of past events and characters, indicating a need to look carefully and
independently at codified history when forming judgements about earlier times.

Therefore,
what role have historians played in the story of temperament?More to the point, what role did people such as Pythagoras play?Most people think they know how pivotal his contributions were – if
indeed they were.But if historians
get things wrong how, if at all, do they attempt to put them right?Indeed, what is history, and how should we regard it?How should we regard historians, who seem to be able to get away with
scholarly blue murder when academics in most other disciplines would be rightly
ridiculed by their peers?All these
questions are pertinent to the consideration of the ancient subject of
temperament, because I am going to tell the story in a way which leaves much of
its conventional history looking a bit skewed.I am going to adopt the straightforward, though apparently unusual,
approach of looking at the facts rather than myths, legends and received wisdom
to see what they really tell us about temperament.

Pythagoras
is widely quoted by Temperamentalists, among whom are a good number of
historians, to the extent that people might be forgiven for believing he
invented the whole subject of tuning and temperament.After all, what paper on temperament would be complete without a mention
of the Pythagorean Comma (perhaps better called the Pythagorean Coma in view of
the boredom it induces among the much-lectured).This article is no exception, but I shall mention it only once more a
little later on – promise (although you might find that promises can sometimes
be broken, even by me).But
when we consider that Pythagoras probably had little or no connection even with
the origins of his eponymous triangle Theorem, let alone with temperament, we
have to consider what history itself means and how we should approach it.Without an understanding of how historians approach their calling, we
cannot understand what temperament really means today, why so many people remain
interested in it, and how to understand what it is really all about.

Pythagoras
lived in the 6th century BCE (I use the term CE -Common Era - here to avoid a potential conflict with an unnecessary
overlay of Christianity).Most will
remember his Theorem from their schooldays, which states that triangles with
sides of lengths 3, 4 and 5 units (or 5, 12 and 13, as well as others) are
right-angled.These values also
satisfy the equation which says that the square of the largest number equals the
sum of the squares of the other two.It
is important to realise that you do not have to know about the equation before
you can understand empirically the right angled property of the triangles (some
historians do not seem to have twigged this - Pythagoras’s Theorem is actually
two theorems in one).In other
words, the Theorem could have existed in an empirical and practical form long
before algebra had been invented, and it almost certainly did.

How
Pythagoras has come to be known as the discoverer of this truism eludes me.A mere couple of centuries later, Euclid does not mention him by name
when treating the subject [1]. One reason for
this was probably the extreme secrecy and isolation in which Pythagoras
worked. Outside a carefully-chosen few known as the Pythagorean
Brotherhood, nobody knew of his discoveries and achievements. Members of the
Brotherhood were sworn to secrecy, so much so that at least one of them was
executed (by drowning) for breaking his oath. Charming feller, wasn't he,
dear old Pythagoras?So
how, more than two millennia after Euclid, can we associate Pythagoras so
definitely with “his” Theorem today?Even
if we were to grant, on the basis of non-existent data, that Pythagoras was one
of several who discovered independently this property of three particular
integers rather than merely retailing it to his disciples, there is now evidence
that others had beaten him to it by a long chalk.For example, it is described on a Babylonian tablet of c. 1800 BCE, and
other evidence suggests that Chinese and Indian cultures of about the same time
were also aware of it.In fact,
significant involvement with the problems of constructing reasonable-looking
rectangular buildings, pyramids, etc must have led to an empirical understanding
of the rule almost as night follows day – there is a high probability that it
would have dawned in the mind of an experienced architect in any civilisation.If you can’t lay out right angles on your building site you
can’t really call yourself a builder.

So
if Pythagoras was not the first to discover the Theorem, or maybe not even among
those who did, are we entitled to enquire why he is also credited with so much
to do with temperament?Yes, we are, because the reasons have as much to do with myth
and legend as anything else.One of
the stories is that he, and only he, discovered the natural harmonic series
merely because he happened to hear a blacksmith hammering bits of metal.Well, I ask you .....The
fact is that, like the Theorem, the harmonic structure of musical sounds was
known to mankind long before Pythagoras (we know this from artefacts such as
ancient Chinese flutes).But even
ignoring the flutes, it’s pretty insulting to our forbears, when you think
about it, to believe that they knew nothing about right-angled triangles and
harmonics in music for the untold millennia before Pythagoras just happened
along.Just like triangles, the notion of harmonics in musical
sounds is something which any number of even half-perceptive individuals must
have pondered on since man first walked the earth.

Why
is this?Because you can hear them!Maybe historians can’t – I could not say - but most other people can.If you blow a pipe you can hear the octave above the fundamental note.The twelfth is even more obvious.As
we are organists on this website, try wedging down a note around the middle of
an organ keyboard with a pencil (carefully!) and then draw the Open Diapason.Walk slowly round the building.If
you can’t hear the twelfth strongly in certain positions (where it is
augmented by its own standing waves) I would be surprised.By blowing a pipe harder you can hear the harmonics even more clearly.Indeed, an open pipe will eventually overblow to the octave, and a closed
one to the twelfth.A child could,
and probably often did, do this, sitting in that incomparable Mediterranean
sunshine all those centuries ago when there was so little else to do all day.A few of those children might have come to realise they had heard
something remarkable in later life.That’s
how many discoveries have happened.

Beyond
the twelfth the harmonics become more difficult to detect by ear.But from the fundamental and these two harmonics above it, just these
three notes, the entire structure of music and (particularly) Western harmony
can be derived.It is a
tremendous vista, a wonderful thing, and the reason why any musician worth the
name needs to understand it.From
just three notes has sprung our entire human musical experience, maybe
mankind’s prime example of cultural unity in diversity.The two harmonics are the octave and an octave-plus-a-fifth (i.e. a
twelfth) above the fundamental.In
fact the fifth is so dominant both in its sound and in the theory of tuning that
it became known as the dominant, and so it remains in today’s nomenclature.It was no great step beyond this to realise that exactly the same
frequencies could be reproduced at will by giving an instrument several strings
or pipes, tuned in octaves and fifths.When
correctly tuned, the music they produced would sound the most euphonious and
concordant – in fact, “in tune”.What
does “tuning” actually mean?In
those days it meant adjusting the tensions of two strings or the lengths of two
pipes until there was no beat or wavering when they were sounded simultaneously.This was true whether the two notes in question were octaves
or fifths (see Appendix 1 for an explanation of tuning using beats).

As
time moved on it was realised that the gap between two notes an octave apart
could be filled with eleven other notes, twelve if you count the bottom note as
well.This too was probably
discovered largely by serendipity, rather than being seen as a remarkable
intellectual leap as some historians claim.Pythagoras is again credited by some of them with much of the clever
thinking here and (as a deservedly famous mathematician) it is possible he did
contribute to the outcome, or reinvent some of it.But he was not the first, because those Chinese had almost certainly
beaten him to it again.It could
have arisen as a result of adding more and more strings to an instrument, all
tuned in octaves and (particularly) fifths of each other.Be that as it may, why this magic number twelve?Because the twelve notes are all fifths of each other.Fifths again!

We
still use these self-same twelve notes in the octave today.To see this for yourself, sit at the piano and, starting at bottom C,
ascend the keyboard by fifths.Thus
you will play the following sequence of notes before you reach another C [11]:

C,
G, D, A, E, B, F#, C#, G#, D#, A#, F, and then top C

Rearranging
them you will find – surprise, surprise -that you have played all twelve semitones of the octave:

C
(twice), C#, D, D#, E, F, F#, G, G#, A, A#, B

You
will get the same result by starting on any other note and ascending by fifths
until you get back to it again – the starting note does not have to be C.

No
wonder the fifth was regarded as dominant!It can generate all the notes of the chromatic scale we use today,
including the octave, from any starting note.Thus the fifth is the generating interval which has created all melodic
musical instruments and therefore all of music itself.In a sense, if you play a fifth you have played all music ever written.
In recent times a mathematician was so intrigued by the properties of
Mandelbrot’s fractals that he used them to regenerate an entire Beethoven
symphony from its fractal representation.What
a pity he apparently didn’t know he could have done this just by playing a
fifth.

This
spine-tingling, amazing discovery (and it is amazing still when it first dawns
on you, however it might have arisen), this seemingly magical property of the
fifth to generate all other notes, is one of those transcendental, golden
achievements of mankind’s collective intellect which reinforces our belief
that the universe is fundamentally simple in the sense it can be understood.In the case of the Greeks it also led to all that Byzantine mythical
mumbo-jumbo which usually clouds discussions of temperament, such as the divine
ratio of 2 to 3, the music of the spheres, the purity of integers, the
relationship of music to the gods, etc.Although
all of this did indeed exist at the time, it can be largely swept aside for
present purposes and left to the historians and the classicists to pick at.But one problem could not, and cannot today, be ignored.This concerns tuning purity.

If
all the fifths are tuned true or pure, in other words if you tune them so the
two notes in each fifth do not create a wavering beat, the topmost C in the
piano experiment above will be significantly out of tune with the bottom C you
started from.It will be painfully
sharp.To get it back into tune,
you have to flatten some or all of the intervening fifths just a little, so that
when you reach top C it is exactly in tune – exactly seven octaves higher in
fact - with bottom C.Therefore,
those fifths which were flattened are now impure – you can hear a beat between
the two notes.The difference
between the out-of-tune top C and the in-tune top C is called the Pythagorean
Comma.Having arrived once again at
this orgasmic climax of the discussion I promise not mention it again (maybe),
together with other unhelpful terms such as Lesser Diesis, Syntonic Comma,
Schisma and Greater Diesis.Together
with somebody else called Didymus whom we’ve never heard of and don’t want
to either.Those Temperamentalists
who delight in immersing themselves in such argot today can nevertheless fall
prey to the most elementary mistakes in arithmetic, some examples of which I
will give later on.Although it’s
funny to see them tripping themselves up, it’s actually pretty serious.

The
ancients strongly disliked this tuning impurity of the fifths because it went
against their mystic ideas. If you believed that your favourite god was up there on the
nearest hill listening to the music you were playing, you were going to be jolly
careful that you didn’t offend him or her.The last thing you would want to do would be to deliberately
choose some hideous irrational number to represent the frequency ratio of a
flattened fifth, rather than the primitive integral beauty of 3 to 2 for a
perfect one (the fundamental frequencies of the notes forming a purely tuned
fifth – no beats - are in the exact ratio of 3 to 2).So they decided to make music using fewer than twelve notes
to the octave, even though they knew all about the twelve note scale.Hence those boring and funereal old modes which characterised music until
about a millennium ago, at least in Europe.The restriction to 6 or 8 notes to the octave made modal music-making
pretty simple during all that time.

But
for some reason the Europeans, uniquely and awkwardly among all musical
cultures, suddenly started to invent the rules of harmony – they wanted to be
able to play several different notes simultaneously, and they wanted less and
less restrictions to be placed on the notes which could be used.So they dusted off the old idea of a twelve note scale.Instead of modes, they also invented the concept of key along with that
of harmony, and this meant that they wanted to be able to play chords in any key
they chose.This brought them up
against the old problem of how to tune the twelve notes of the octave so that
all keys could be used, which is the root problem of temperament.“Tempering” the fifths, or flattening some or all of them so they
will all fit into the octave, is what temperament is all about. And having
approached the subject in forward time, we shall shortly jump to the present day
and look backwards.But before
doing that it is worth saying a few words about the truly great gift which the
ancient Greeks, Pythagoras included, gave to us.

We
have already touched on the mad antics of their gods and the notions of purity
and correctness which pervaded their culture.Yet at the same time the gods did not seem to mind the Greeks indulging
in any amount of philosophy about life, the world and everything.This laudably liberal attitude on the part of their otherwise
idiosyncratic deities was remarkably unlike the attitude of the Christian Church
and some other religions closer to our time.So we find that the notion of philosophy itself was born around the time
Pythagoras lived.It could be said
that philosophy was invented by Thales of Miletus in Greece.He taught that the Earth was a flat disc floating in an infinite sea, and
that everything arose from this watery medium.What is significant about this is that it was a theory which
offered an explanation, however bizarre we might think it today, for the
otherwise mysterious and inexplicable things which we all perceive as part of
our being and of living.

Thales’s
idea was not a religion, relying on divinely revealed wisdom.It was purely an intellectual construct of the mind which could be tested
and, if found to be wanting, it could be discarded.In parallel, the work of mathematicians such as Pythagoras and (later)
those with a more experimental bent, such as Archimedes in nearby Sicily,
allowed this and other theories to be tested against observable facts, against
discovered truth as that polymathic thespian Stephen Fry called it in a
memorable lecture to young minds a few years ago [2].The Greeks realised that if a theory was disproved by just one
observation, it was disproved for ever.It
could no longer exist.Conversely,
a theory could never be proved either.It
would remain on the stocks just as long as it wasn’t disproved.These fundamental tenets of philosophy and science have remained
unchanged ever since, and they have stood us in good stead.

Thus,
during a golden age lasting only some 500 years, the Greeks and their immediate
successors laid the foundations of our modern systems of thought which are still
used unchanged today, a tremendous legacy of intellectual freedom which later
threatened to undermine those who saw their future in a different direction.Among these was the Church.For
about the first 1500 years CE it displaced the notion of discovery by priestly
revelation, and it was not until this stranglehold began to disintegrate a few
centuries ago that the next phase in the story of temperament really began.

Before
proceeding, what have we covered so far?

1.A single musical note has a natural harmonic structure which can be
heard.

2.The first two harmonics above the fundamental note, the octave and the
twelfth, can be given a separate existence by tuning separate strings or pipes
to the same frequencies.

3.The twelfth is particularly important because it is a fifth above the
octave.

4.Using only the interval of a fifth, the octave can be spanned by twelve
different notes, all generated by fifths.

5.But if derived from purely-tuned fifths, these notes do not quite fit
exactly into an octave.

6.
Therefore some or all of them have to be squeezed or flattened.

7.The choice of which fifths are flattened, and by how much, is the study
of temperament.

In
the previous section we got roughly as far as the 16th century by
travelling rapidly forwards in time from the era of the ancient Greeks (nothing
much happened for the first 1500 years or so after BCE became CE, so we skipped
over it).Now we shall look back
towards the same period from today’s vantage point.

Why
look backwards?Because the study
of temperament today is almost invariably backward-looking.What Temperamentalist writes about anything other than the temperaments
allegedly used by Bach or his contemporaries, for example?More than a few Temperamentalists seem to want to make a name for
themselves by propounding increasingly fantastic hypotheses about subjects such
as the temperaments favoured by Bach.It
is fertile ground for those with the inclination, simply because much of what
they claim is, and will forever remain, unproveable.The big problem in temperamental retrospection of this kind
is that it almost invariably overlooks the differences between life today and
life three hundred or so years ago.Thus
the work is by definition anachronistic.It
is out of time, studies not so much of temperament but examples of reverse
anachronism.This sort of
Temperamentalist transplants today’s understanding, culture, values and
capabilities several centuries backwards as though these things have not changed
in the intervening period.Reading
some of this material suggests to me that the authors imagine that 17th
or 18th century theorists, musicians and composers understood the
physics of music just as we do today, that they had electronic tuning aids, and that they had no difficulty in
performing calculations of the most involved sort as though electronic
calculators were the norm.Such
assumptions are of course grossly false.These authors either do not know their history at all, or
they have only absorbed that regrettably skewed version of it which ignores the
awesome leaps made in human understanding over this period.So let us have a peek at what life in Europe was really like at the end
of the 17th century when Bach was becoming active as a musician and
composer.

A
big issue was that life was still pretty much dominated by the Church.In Bach’s Germany and in Britain that meant Protestantism. And it
wasn’t the mellow experience we get today when we go to church, either.The Protestants had not long stopped burning Roman Catholics in village
squares, leaving those who lived in nearby houses to scrape the human grease off
their walls before they became alive with maggots.Also witchcraft was an ever-present fear among the population, with the
last documented witch execution still to take place in Germany in the 1770’s,
over twenty years after Bach’s death. Officially-sanctioned torture of
suspected witches was not abolished in Bavaria until the early1800’s.

In
such times you would therefore have been well advised not to get on the wrong
side of the Church or the civic dignitaries. Although
the political clout of the Church at national level was on the wane, the local
clergy could still wield a lot of influence when a young man wanted a job,
particularly a job as a church or Court musician.Physically trapped within the insularities of village life in an age
without means of transport beyond your own legs or the horse, you would
generally not want to attract attention by doing anything out of the ordinary,
such as being seen to take an excessive interest in anything at all. Made
wary by the religious wars which raged in Europe in the 17th century, even
Descartes said that "to live you must live unseen". Therefore,
dabbling in science (an anachronistic term in itself because the word did
not gain its current meaning until the 1830’s) would have been a definite
no-no.By and large, the ancient
Greek concepts of unfettered intellectual freedom had been forgotten for more
than a millennium because of religious suppression of the most vicious kind.Isaac Newton did not go up to Cambridge until 1661, only 24 years before
Bach was born, and even he had trouble with the university because he was
reluctant to accept the Doctrine of the Trinity.What was not to be found in the Bible was potentially dangerous,
associated only with reactionaries.

So
are the Temperamentalists really expecting us to believe that organ builders,
organists and composers routinely sat up nights at a window by the light of a
guttering candle doing sums about tuning?Would
that have fitted with the contemporary notions of good citizenship?Was there a danger that it could all too easily have been misinterpreted,
maybe as witchcraft, if somebody else caught sight of all that spidery scrawl
and countless numbers?Page after
page after page of it would have been necessary to develop a new temperament.
If floods or other pestilences descended on a village, would not such people
have been regarded as architects of these misfortunes? I do not know the answers to these questions, though there is a certain
plausibility about a scenario which suggests that on the whole they would not
have done these things.But I do
know that, apart from a minority, they could not and would not have done them
for other, very good, reasons.That
minority would either have been extraordinarily determined, or it would have
possessed knowledge and capabilities well beyond the average, or both.Let us look first at what was known about what we would call the physics
of music (though they would certainly not have called it that, again because our
term ‘physics’ was not defined until the 1830’s).

At
the most basic level, we can ask whether there was even a general awareness in
those days of the concept of frequency, in the sense that a string vibrating a
certain number of times per second would emit a note of a certain pitch (we
ignore here the difference between objective frequency and subjective pitch).Of course, it would have been obvious that a string did vibrate
transversely while it was emitting sound, because of the tizz it made if you
brought your finger nail up against it.But
was there any realisation that a string vibrating at middle C vibrated with
exactly twice as many oscillations per second as when it was sounding tenor C?If so, how would they have known this for sure?This is not mere pedantry, because exactitude necessarily lies at the
heart of tuning and temperament as we shall see.There was no possible means of rendering the detailed motion of the
string visible, nor of measuring frequency.The numerical exactitude required in temperament studies is around 1 part
in 100,000 or 0.001% (see Appendix 2).

Note
that I said "general awareness of the concept of frequency"
above, meaning that the majority of musicians would probably not have given
these matters a second thought. This does not mean that the knowledge did
not exist at all, because some texts from the mid-17th century mention it [14]
and I am grateful to the harpsichord maker J-P
Baconnet for suggesting I point this out [15].
But my point remains : it stretches credulity to believe that these matters
would have been taught routinely as part of the general education of the day,
whatever that might mean. Even in Britain before the second world war
hardly any science was taught in the State school system as my late father often
reminded me, and he had won a scholarship to a relatively élite school!

Notwithstanding
all this, musical instruments still had to be tuned, and tuning
at that time would have been carried out by counting beats as it often is
today.No problem there then,
provided you had a source of portable time. By this I signify some means
of measuring beat rates at the keyboard, expressed as (for example) 5 beats in 7
seconds. But what form would such a source of portable time have
taken? A wristwatch? Hardly, considering they did not become
available until the end of the 19th century. What we would call pocket watches
had been around in some form since the 1520's but they were generally crude, expensive and
relatively rare. By no means all of them had the necessary sweep seconds
hand, or indeed any form of seconds hand. That sort of pocket timepiece had to await the mind-blowing work
of Harrison who did not produce his quite exquisitely beautiful pocket chronometer until the
1760's, by which date Bach had died.

At
the time we are considering it is more likely that most clocks used pendulums
and were therefore not portable. The
timekeeping property of a pendulum had been discovered by Galileo around 1580, but it was not until Newton
had explained fully how it worked that the pendulum escapement became a domestic
commonplace a century later - around the time Bach was born - in timepieces such
as the long case (Grandfather) clock in Britain. Some of these would have measured the
passage of time audibly with a "seconds pendulum" whose ponderous
tick-tocks were a
second apart, and there is a chance you might have developed an innate mental capability to
estimate seconds reasonably accurately if you had grown up in a house
containing such a clock. However, my favourite conjecture is that
musicians might have carried an elementary pendulum around with them, thereby
solving the problem of a source of portable time for tuning and maybe for
setting the pace of music as well. (Maelzel's metronome did not appear until
1815. Until the advent of today's electronic devices, cheap plastic pocket metronomes
using the pendulum principle were readily available from music shops almost up
to the present day. My first teacher in the 1960's used one frequently in
the organ loft). Such a musician's pendulum in the 17th century or so could have been identical
to the plumb line which had been used by architects and builders for countless
millennia, consisting merely of a piece of string with a weight at one
end. The string could easily have been calibrated with marks
indicating the beat rates of the fifths of your favourite temperament [12].

There
is also another aspect of the matter. If measuring beat rates was in fact
regarded as more difficult than I have assumed here, then, obviously, the more pure fifths in a
temperament the better from the point of view of ease of tuning. It is much
easier to tune a fifth pure than to temper it, because when in tune it has no beat.
The unequal temperaments which sprang from the period we are discussing
frequently had many pure fifths, such as Werckmeister III (c. 1690 - 8 pure
fifths out of 12), Vallotti (c. 1730 - 6 pure fifths out of 12), etc.
Setting such temperaments by ear, especially for an amateur tuner, is much
easier than setting Equal Temperament, because ET contains no perfect fifths
and all twelve have to be accurately tempered. Anyone who disagrees will
need to explain why professional tuners today frequently do not tune by ear but
instead use electronic devices when tuning organs or pianos.
Maybe this humdrum matter contributed to a preference for unequal temperaments
until instruments with better tuning stability were developed in the 19th
century, an issue to which I shall return presently.

But
returning to the physics of music, was it understood what a “beat” was, in the quantitative sense of
it being a measure of frequency difference?And a difference of which frequencies exactly?How many Temperamentalists even today understand why the interval of a
fifth creates a beat when it is not exactly pure?The fundamental frequencies are grossly different even when the interval
has been tuned pure so that there remains no beat – the frequencies of a pure
fifth are in the exact ratio 3 : 2.Why
is this?How can there be no beat
when there remains such a large frequency difference, yet why does a beat appear
when this massive difference is made a tiny amount smaller or larger?In a recent paper Lehman attempted an explanation by stating that the
“upper harmonics of both notes have a frequency that is almost identical, at
some point several octaves above the fundamental” [3].This statement is incorrect.Clearly,
then, these are matters which tax the understanding even of some specialist
writers today, so are Temperamentalists telling us that such knowledge was
widespread among the general musical population in Bach’s day?(The answers to the questions just posed are in Appendix
1).

Then
there was the sheer problem of doing arithmetic.We have already noted that the arithmetic involved in tuning and
temperament calculations has to be done to a precision of at least 1 part in
100,000.The necessity for this is
explained in Appendix 2.In other
words, the calculations have to be done using six significant figures.How would multiplication and division of two 6-figure numbers have been
done in those days?By long
multiplication and division of course.It
hardly seems necessary to add that calculators were unavailable!Logarithms had already been invented, and these could in principle have
eased the task somewhat.Although
log and antilog tables were available (though how widely available in your
average village I do not know) they were full of errors, literally thousand upon
thousand of errors, which is one reason why ships often had no idea where they
were (the other main reason was the impossibility of keeping accurate time on
board).Were boys such as the young
Werckmeister and the young Bach taught logarithmic methods of doing arithmetic?Even if they were, with the errors in the tables it would have been a
pretty fruitless endeavour as far as temperament was concerned.Various types of slide rule had recently (c. 1660) been introduced, based
on logarithmic methods of doing arithmetic, but the precision offered was not
sufficient for the necessary calculations involved in nuances of tuning.A vast variety of “ready reckoner” tables was later published, but
not until the 18th century was well advanced and by then Bach had
died.Even then, they were riddled
with errors just like the earlier log tables.In fact the problem of error-ridden typography was the main driving force
behind Charles Babbage’s unsuccessful endeavours in the 19th century to produce
a mechanical computer – one of its key features, on which Babbage insisted,
was that it had to be able to print its results automatically and thus not rely
on a human type setter and proof reader.

Let’s
look at an example of Baroque arithmetical problems.The Pythagorean Comma (oops – I’ve broken my promise – here it is
again) can be represented as the ratio of the two repellent numbers 531441 and
524288.Notice they both have those
six figures which we mentioned earlier.Although
these numbers would have been known to the clever minority we spoke of above,
how would this quotient – the answer to the division - have been arrived at in
the 17th or 18th century?Such a calculation typifies those involved in the arithmetical study of
temperament.Still, having done
such arithmetic at junior school in the 1950’s, I know it can be done and how
to do it.I also know how error
prone the results can be, and a rap on the knuckles with a ruler was not unknown
in those days if you got your sums wrong too often.(How things have changed, and for the better!).The method of long division by repeated subtraction is demonstrated in
Appendix 3 for the benefit of those who are curious to see how sums were done in
pre-calculator days.I do, however,
question whether doing such sums was a favourite way of whiling away the
evenings in the 17th century.Even
with the benefits of today’s computers, calculating aids and computerised
typesetting, modern Temperamentalist literature is (disgracefully) still riddled
with numerical errors, examples of which I shall demonstrate presently.What it was like then can well be imagined.

Yet
the extraordinary paradox is that musicians in those days probably had a better
intuitive feel for tuning and temperament than their brethren do today.This is largely due to the single fact that the stringed keyboard
instruments used for general music making, such as the harpsichord, had wood
frames.In fact the harpsichord
case itself had to withstand the combined pull of the many iron and brass
strings – there was no frame at all as such - and the fact it sometimes did
not succeed is shown by some surviving examples where the case has virtually
imploded.Today’s piano has a
massive cast iron or welded steel frame (early ones did not) to resist the
tension of the strings, which is why it is so heavy.The rigidity of the frame endows it with a tuning stability that people
would have given their eye teeth for three hundred years ago.Music can be played quite happily on a modern piano, even a
cheap upright, which is tuned only once or twice a year, whereas the tuning of
instruments with wood frames was much less stable.Tuning might have been required several times a week in some
cases, and the only way to do it was for virtually every village player to learn
how to tune as part of his or her music lessons, just as today’s string
players still have to do.

I
cannot believe that tuning would have been regarded as anything other than a
chore, to be undertaken before the day’s music making started perhaps.It would have been carried out as quickly as possible, and doubtless it
was often slapdash to some extent.Often
I imagine it would have been skipped altogether.Thus tuning errors would have abounded, the only criterion of whether a
tuning exercise had been successful being whether the latter state was better in
some sense than the former.But,
surely, herein lies the clue we need to understand why so many temperaments
discussed today sprang from that period.The
argument has two strands: firstly, we do not need to waste time discussing here
whether “good” temperaments were sought because they obviously were.By a “good” temperament I mean one in which most if not
all keys can be used without gross dissonances.Bach’s ‘48’ is probably the best known example demonstrating the
hunger for instruments which can be played in all keys.But please, please bear in mind that there are lots of
“good” temperaments.The term
does not solely mean Equal Temperament, and in any case some today would deny
that ET is “good” anyway.Maybe
that was true then as well.I do
not intend to get dragged down this side alley here.Consult the writings of almost any Temperamentalist for more on this.

The
second strand to the argument is that, because of the probably slapdash tuning
or its complete omission, occasional happy coincidences would arise in which a
particularly euphonious result was achieved in a particular piece of music.Yesterday it might have sounded terrible, but today it sounds much better
because of the hurried tuning done this morning, or because we didn’t bother
to tune the damn thing at all and it just happened to have been very cold last
night.Noting the felicitous result
with the piece we have just been practicing, a few other pieces in other keys
would quickly be played.If the
results remained good, someone with the sharp wit and intellectual mightiness of
Bach could well have paused to write the tuning down.And how would he have done this?Just
like today, he would probably have run through all the fifths and fourths
(fourths are inverted fifths) over the middle octave or two or three, counting
the number of beats in a given time interval such as ten seconds, and writing
them down.Or he might have used
some form of shorthand to indicate whether they were flatter or sharper than
those which had actually been intended when he last tuned the instrument (though
having counted the beats anyway, I suspect he would have simply have written the
numbers down unless there was a need for covertness).

I
have had the good fortune to discover a “good” temperament this way, merely
by playing a village organ which by chance had not been tuned for a while.I called it “The Dorset Temperament”, and both it and the experience
which led to it are described elsewhere in the Temperamentalist literature and
on this website [4].As a
physicist, I hope you will not regard me as immodest if I claim some familiarity
with mathematics and number manipulation.Yet,
like most other people I suspect, the thought of trying to sit down and design a
new temperament from scratch by doing sums seems not only repulsive, but
completely the wrong way round to do the job from an intuitive, musical,point of view.I suggest most erstwhile Temperamentalists three hundred
years ago would have felt the same, but even more so.

If
this argument is accepted it means that a number, maybe the majority, or even
all “good” temperaments would likely have arisen largely through
serendipity.It is just too
implausible to believe that the inventor of a temperament first sat down and did
lots of sums before trying it out for real.In reality it would have been the other way round – the
temperament appeared one day as if by magic, and it was then analysed
retrospectively by those with the inclination.This does not mean that an interesting tuning arrangement which arose by
chance would not have been tweaked subsequently after doing a few sums.But lest this seems to denigrate the capabilities of those who have
donated their names to their eponymous temperaments, bear in mind that much of
today’s science is still done this way.Something
is observed, maybe by chance (such as the discovery of the bactericidal action
of a mould which turned out to be Penicillium), and the explanation is sought
later.

The
same thing happens routinely in the natural sciences such as physics –
experimentalists do an experiment which occasionally might have a completely
unexpected outcome, and then the theoreticians get going to explain why.Seldom do the processes of science take place the other way round (one of
the few counter-examples being the theory of black holes in cosmology, where the
theory was developed in detail over about two hundred years before the
observational evidence of recent times was obtained).Why should this not have been the way in which “good” temperaments
were discovered – serendipity followed by theory?Alexander Fleming, the discoverer of penicillin, said some memorable
things when looking back to that day in 1928 when some mould or other just
floated through the window of his laboratory.Two of them are“if my
mind had not been in a reasonably perceptive state I would not have paid any
attention to it”, and “before you can notice any strange happening you have
got to be a good workman, you have got to be a master of your craft” [10].Nothing better epitomises my view of the serendipitous way
“good” temperaments arose than these words:they could just as easily have been written by Bach himself.

There
is an amusing sequel to this story which might have relevance for historians
studying temperament.When writing
up the results of their work, some scientists find it difficult to admit that
they stumbled across their results by accident, so they write their paper as
though they were terribly clever and worked up the theory first, testing it by
experiment afterwards.Some modern
historians seem to take much 17th and 18th century work on
temperament at face value – I feel they do not analyse the situation deeply
enough, and with sufficient understanding of the real process of discovery.It is generally more plausible to accept the
serendipity-followed-by-theory sequence rather than the reverse, despite what
the old texts might imply.

Interestingly,
there can be no argument that an exact reversal of the hypothesis (that
instruments with unstable tuning led to the discovery of "good"
temperaments) occurred later on. In the 19th century the universality of
Equal Temperament came about precisely because of the dominance of keyboard
instruments whose tuning was extremely stable. These were the free
reed instruments such as the concertina, the accordion, the harmonium and the
American Organ. Unlike organ pipes, the tuning of free reeds is virtually
unaffected by temperature and wind pressure, indeed they were often used as
frequency standards in Victorian laboratories. The rapid spread of such
instruments throughout the world was facilitated by the various European empires
which then existed, and therefore the instruments took Equal Temperament along
with them. Both examples, that is, instruments whose tuning is either
stable or unstable, resulted in the propagation of different sorts of
temperaments quite independently of what the theorists of the day might have
said or thought. On the whole musicians will simply play and compose for
whatever instruments happen to be around, and their music reflects this.
They seem to take little notice of the Temperamentalists wittering away in the
background.
Whatever our views about Equal Temperament, we can be grateful for the music it
has given us from the 19th century.

However,
we digress. Reverting to unstable tuning and going
a bit further, a by-product of the serendipity process might have been that some
“good” temperaments included impure octaves.It is unclear to me why the octaves should be any more sacrosanct in the
regime of controlled detuning which is temperament than any other interval.Yet what well known temperament today actually uses impure octaves?If the scenario of frequent and slapdash tunings which I have postulated
is correct, it is likely that euphonious results were occasionally obtained in
which at least some of the octaves were impure – and most likely sharp.I shall return to this possibility later.

So
at the end of part two, what are the main conclusions?

1.Modern temperament studies are necessarily retrospective, but often with
insufficient heed paid to the circumstances of life at the time studied.In this sense they can be anachronistic.

2.The influence of the Church would have made people more wary of indulging
in free thought and research than we take for granted today.

3.In earlier times there were not the arithmetical tools available, nor the
depth of knowledge about the physics of music, to enable the quantitative
analysis of temperaments to be made as readily as today.

4.Nevertheless there was probably a better and more widespread qualitative
feel for temperament and tuning issues than today, largely because of the
notorious tuning instability of stringed keyboard instruments which required
most players to be able to tune them.

5.The need for frequent retuning might have led to the serendipitous
discovery of some or all of the temperaments we associate with that era.

6.Routinely developing a new temperament by the reverse process of first
doing the theoretical work would probably have been rare, despite the
implications contained in some contemporary publications from the period.

Apart
from the issue of reverse anachronism, it is worth taking a wider look at some
modern work on temperament. It seems to be one of those subjects which
attracts more than its fair share of armchair experts if the contents of Internet
chat lists is anything to go by. These have done no actual work in the subject at all in the
sense of having published anything. This means they do not distinguish
between erudition and scholarship, and a large proportion of them also believe that
invective is an acceptable substitute for either.

But
of the work which is published, probably
the most widespread and least excusable matter is that too much of it is peppered with arithmetical errors.
For
example, Kent and his publisher (CUP) apparently believe that 1200
x 2 + 386.14
equals 2786.314 [5].Like many
other authors he also draws heavily on data from Padgham’s treatise on organ
tuning [6], and in view of the errors in the latter it is to be hoped that he
verified them independently.For
example, there is no way that the Werckmeister III temperament could be set up
using Padgham’s numbers.Similarly,
Padgham’s table on Just Intonation contains a misprint that should have been
obvious to even the most casual proof reader.Throughout his book are innumerable examples of varying arithmetical
precision, revealed by unexplained (indeed, inexplicable) differences in the
numbers of significant digits.You
cannot do meaningful temperament research unless you work to at least 6
significant figures throughout (see Appendix 2 to this article).

Continuing,
Lehman [3] states that “two consecutive 5ths of equal size ... have beat rates
in a 3:2 ratio”.He quotes the example of the interval G-D which includes
middle C and the interval D-A above it, though "adjacent" might have
been a better adjective here than "consecutive" (which has confusing associations
with harmony). The meaning of the terms "beat
rate" and "equal size" is also unclear. For the former,
perhaps Lehman meant "beat frequency" and we shall assume here that he
did. As for the latter, if these two "equal size" intervals were tuned to
Equal Temperament (in which they are both equally flattened from pure) the ratio of their beat
frequencies would be 1.49849, which differs
from his value by slightly more than 0.1%. Such differences are highly
significant in the context of tuning and temperament, where we have already seen
that the difference between two numbers can only be ignored if it is less than
0.001%, one hundred times smaller than the percentage error here. If both “equal size”
intervals were to be tuned pure, which they might be in a number of actual
temperaments, there are no beats at all.This
would lead to a ratio of 0 divided by 0 which is a mathematically indeterminate
quantity, very different to Lehman’s 3:2. Thus we have been badly cast
adrift here in an arithmetical sense.

These
are just a few examples.I do not have the time nor the inclination to act as honorary
proof reader for all of the myriad publications I consult when doing theoretical
work on temperament, thus they are merely some of the examples I have come
across by chance. More than once I have wasted hours tracking down arithmetical
problems which turned out not to be mine (my first thought) but due to errors in
published data, including one of the cases above.The examples quoted are all relatively recent, and two of them are scarcely from what one would normally regard as
twopenny-halfpenny publishers.So if the quality of the data is as bad as it obviously is, what is the
point of publishing it?Do the
authors themselves actually use their own numbers for anything beyond conflating
their publication tally, or is it perhaps a case of “see how clever I am”?If the latter, it is in reality a case of “see how good I am at digging
traps for myself” I’m afraid.

I
have mentioned earlier the effort which goes into trying to prove which was
Bach’s favourite temperament.One of the most memorable (to my mind) papers was published
quite recently.Entitled
“Bach’s extraordinary temperament – our Rosetta Stone” [3], I have
already cited two erroneous statements from it.

But
the main thrust of the paper is that the cypher used by Bach on the title page
of his Das Wohltemperirte Clavier (above) is no mere squiggle but encodes the
temperament he wanted to be used when playing these pieces.Despite the inconvenient facts that the squiggle apparently has to be
turned upside down before it can be analysed, and that no comparable method of
encoding temperament information was apparently used again by Bach or anybody
else, the author claims that its meaning was so obvious that it was used by the
Leipzig authorities when assessing Bach in his absence for a teaching position
as a successor to Kuhnau.So
obvious, in fact, that they would have tuned a harpsichord to this temperament
before trying out his pieces in all keys which were part of his submitted CV for
the job. Unfortunately not a shred of collateral factual evidence is presented
to support the hypothesis though.Surely
if Bach wanted to encourage the use of a particular temperament when playing
these pieces he would have set out its characteristics in a less equivocal
manner?I wonder if the author was
conscious of the inevitable comparison he invites by his choice of title between
his capabilities in decryption and those of Thomas Young and J-F Champollion?

Taking
a more charitable view of the idea there is, however, a further opportunity for
determining the hidden meaning of the cypher which the author of this paper
missed, and that concerns the terminating flourish at the right hand side.It was added deliberately in the sense the pen was first
lifted from the paper before drawing it.If
there really is the amount of information to be extracted that he maintained,
perhaps he might have regarded that flourish as evidence that Bach stretched the
octaves of his temperaments, a possibility not discussed in his paper and one to
which I shall now turn.

As
a rule Temperamentalists ignore or dismiss the possibility that the octaves
might play a greater role in their subject than merely marking the boundaries
between successive sets of twelve tempered semitones.Most of them never even mention it as an option; they proceed as though
pure octaves are axiomatic and always have them tuned true.Inevitably, this leads to a subjective tuning rigidity across the compass
of the keyboard of an instrument, regardless of the temperament to which it is
tuned.The tempering of the
intervals in every octave is the same, and every note is tuned true with its
octaves above and below.The
results are legion.

For
example, the beat rate of any interval played depends on the octave in which the
interval resides.In other words, a
fifth played in the third octave will beat faster than if it is played in the
second octave, but slower than if it were to be played in the fourth octave.With any temperament which uses pure octaves, the ratio of
these beat frequencies has a simple numerical relationship to the octaves considered – a
fifth in the third octave beats exactly twice as fast as when it is played in
the second octave, four times as fast as in the first octave, and so on.These
exact and simple integral beat frequency ratios also apply to any other
interval, no matter how finely they might have been mutually adjusted within
each octave by adopting a particular temperament.With a recently tuned instrument in which all the octaves are
well in tune across the whole keyboard, this can lead to a hard, sterile
locked-up type of sound when chords are played which span a significant part of
the compass.Not only are there no
beats at all between the octaves, but the beat rates between similar intervals
in different octaves are related by exact integer ratios.The sterility only recedes when the tuning of the instrument drifts over
time.The subjective effect can be
even more noticeable and unpleasant in electronic instruments if unintelligently
tuned, because their tuning never drifts once set.Again, remember that the effects we are discussing are independent of the
temperament actually used; they follow purely because the octaves are locked in
frequency.

Subjectively,
the effect is worst for the organ; less so for stringed keyboard instruments.This is because of the different overtone (partial) structures in the two
cases.The overtone frequencies of
organ pipes have an exact integer relationship with each other: the second
harmonic of any pipe, sounding the octave above the fundamental, is at exactly
twice the frequency of the fundamental, the third harmonic (the pure twelfth) is
at exactly three times the fundamental, etc.All the harmonics of an organ pipe are rigidly locked in phase with each
other while it sounds.Using the
terms properly and rigorously, this is why the overtones in this case can be
called harmonics.The situation
described only pertains when the pipes are sounding in their steady state
speaking regime after the attack transients have died away, which of course
occurs relatively quickly.Therefore,
if an organ is well in tune so that the octaves are as exact as possible, the
sterility both of the octaves and of the beat ratios between octaves is
amplified subjectively by the lack of numerical freedom in the frequency ratios
among the harmonics of the pipes themselves.The mere fact that the sounds of organ pipes do not die away until the
keys are released adds yet further to the potential subjective hardness of the
overall effect of an organ with well tuned pure octaves.

Subjectively,
the effect of stringed keyboard instruments does not suffer as badly as that of
the organ even when they are tuned as well as possible.The main reason is that the harmonics of a struck or plucked string (not
a bowed string) are actually not harmonics at all; they are overtones because
their frequencies are not exact integer multiples of each other.As the sound of a struck or plucked string dies away, the overtones beat
with each other because the higher ones are slightly sharp to those lower in
frequency.(This
also happens during the attack and release transient phases of organ pipe
speech, but because these phases are of such short relative duration the effect
is dominated by the steady state phase in which the overtones are true
phase-locked harmonics as described above).

Another
disadvantage following from the use of pure octaves is that an opportunity is
missed to ease the tight straitjacket of conventional temperament work.The root problem of temperament is to squeeze that uncomfortable set of
bedfellows called the semitones into an octave in such a way that none of the
intervals between them is grossly out of tune.This is done by making small adjustments to their frequencies (e.g.
making the fifths flat, the thirds sharp, etc).Why not ease this problem a little by making the octaves
themselves adjustable as well? Sharpened
or “stretched” octaves are nothing new in keyboard music, of course.Many piano tuners routinely stretch the octaves when tuning, though the
reasons quoted vary.Some maintain
that the change in frequency of the partials during the decay of the sound is
less dissonant (when several notes have been keyed) if the octaves are tuned
slightly sharp towards the top of the keyboard.Others say that it is better to tune the octaves sharp so
that they will in time come better into tune as the string tensions relax
slightly.Of the two reasons I
incline preferentially to the common sense nature of the latter.Other reasons for stretching the octaves also exist.At the other extreme though, some tuners will have nothing to do with it
and one well known book on piano tuning does not mention it at all [9].So it seems we can learn little from the piano scenario.

Yet
impure octaves would without doubt have occurred if we accept the notion
described earlier in which “good” Baroque temperaments arose largely through
serendipity because of the tuning instability of the old stringed keyboard
instruments.Therefore I plan to investigate the matter in more detail,
specifically for the organ.Currently
I have yet to decide on a definite road map for the study though.Doing it with the degree of emphasis on arithmetic and theory which
constitutes current work on temperament is almost certainly debarred.It is debarred because pure octaves underpin the entire concept of
temperament as it is understood today, and removing them will also remove the
relative arithmetical simplicity of the subject.If the octaves are no longer pure, the subject could easily become
theoretically anarchic and entirely experiential.Any note on the keyboard could in principle take any frequency value, and
the frequencies actually chosen would then arise solely through empiricism –
trial and error.

To
prevent this unpleasing prospect developing, it will probably be necessary to
impose a deterministic progression of octave sharpening across the keyboard.As an example, a temperament would be set for the lowest
twelve notes, say, then the successive octaves above each one could be sharpened
progressively to generate its upper brethren.Or the generating temperament could be set in an octave closer to the
middle of the compass and then propagated in both directions, up and down,
according to certain rules.Each
octave could contain a completely different temperament in principle, though in
practice it will probably be better to regard the entire keyboard as a
collection of notes upon which the notion of a distributed temperament is to be
imposed. The
availability of computers with electronic keyboards connected to them makes the
execution of such a study not only precise but rapid and interesting, as the
results of any algorithm for distributing the note frequencies can be
immediately assessed by ear.And
that, at the end of the day, is what it is all about.

Why
do we perceive a beat – a wavering effect – when we are trying to bring a
pair of pipes or strings into tune?It is more complicated than most people think, and I have yet
to see a proper explanation of it in print.The following is that proper explanation.

To
begin the explanation, consider a case where two pure tones – sine waves –
are sounding simultaneously.The combined waveform is merely the sum of the two individual
sine waves, so it can be written as:

sin
2π f1 t + sin 2π f2 t

where
f1 andf2are
the frequencies of the two tones in cycles per second or Hertz (abbreviated as
Hz), andt is time in
seconds.π (pi) is the
Greek symbol having the value 3.141593 (to a precision of 7 significant
figures).

The
diagram below is a plot of this equation for frequency values 588.00 and 587.32
Hz.The term SPL stands for Sound
Pressure Level for the sound wave in the air.I’ll explain presently why these apparently outlandish values of
frequency were chosen.

You
can see immediately that there is a slow modulation of the strength of the
sound, which rises to a maximum before falling momentarily to zero.This cycle is repeated, leading to the phenomenon of what we call beats.The beat frequency equals the difference between the two
frequencies f1
andf2 , so in
this case the beat frequency is 0.68 Hz.Hertz
is another name for cycles per second so the reciprocal of this figure, 1.47
seconds per cycle, shows that each beat cycle takes place over 1.47 seconds
which is a little under 1 ½ seconds.This
is a very slow beat, characteristic of the beat rates of the slightly impure
intervals used in temperament work.The
pitch of the note heard is that of treble D (an octave plus a tone above middle
C).

But
in fact this beat waveform is not at all like the beats you actually hear when
tuning organ pipes, and there are two reasons for this.Firstly, the sound of two pipes beating does not actually go through zero
once each beat cycle as it does in the example we see here.This is because the sounds of the two pipes are mixed in the listening
room, and this mixing process is far more complex than simply adding the two
sine waves together which we did here to generate the waveform.The natural mixing process involves a huge number of
additions due to the huge number of reflections in the room, and this results in
a less pronounced beat.Although
there would still be a beat, it would not go through zero because the multiple
reflected sounds would prevent complete cancellation at any point in the beat
cycle.

The
other reason why real beats do not sound like this example is because the sound
of any organ pipe does not consist of just one harmonic – a single sine wave
– but of several harmonics.This
is illustrated below for a Stopped Diapason pipe.

The
Frequency Spectrum of a Stopped Diapason Pipe

The
vertical lines in this diagram indicate the strength in decibels of each
harmonic (you can ignore the fact that decibels are used, and that the harmonic
lines are not equally spaced along the horizontal axis).The fundamental frequency of the pipe is the same as the frequency of its
first harmonic.Successive harmonics lie at successive exact integer
multiples of this frequency i.e. at exactly twice, three times, four times the
fundamental frequency, etc.When
two such pipes are sounded together, you get a separate beat generated between
each harmonic of one pipe and every harmonic of the other, leading potentially
to a large number of beats.However,
because the strengths of the harmonics fall off rapidly, only the beats
generated by the first few harmonics are heard in practice for this type of
pipe.

Tuning
Unisons

When
tuning two pipes which are intended to have the same pitch (e.g. middle C on an
8 foot Open Diapason and an 8 foot Cornopean) you adjust their fundamental
frequencies to be the same.By adjusting their fundamental frequencies, you are also
adjusting all their harmonics at the same time.You simply adjust the frequency of one of the pipes until no
beats are heard, and at that point they are then in tune.

Tuning
Octaves

A
different situation pertains if you want to tune, say, an 8 foot Open Diapason
to a 4 foot Principal.The two are
an octave apart, so even when they are in tune there will still be a large
frequency difference between their fundamental frequencies.At middle C this difference frequency will be about 262 Hz.Therefore, how can you use beats to tune them when there will still
remain this large frequency difference?Does
this not result in a very fast beat even when the two pipes have been brought
into perfect tune?

Because
in this case of two perfectly tuned pipes, the beat frequency between the two
fundamentals is exactly the same as the fundamental frequency of the 8 foot
pipe, and the concept of a beat then becomes meaningless.There can be no beat at a frequency which is the same as one of the
generating frequencies.But if this
is so, can beats still be used in the tuning process, as they were for the more
obvious case of the unisons?Yes they can, and this is how the beats arise.When the pipes are very nearly in tune, there will be only a small
difference between the fundamental frequency of the Principal and the second
harmonic of the Open Diapason, and it is between these two frequencies that the
main beat will be heard. (There will also be subsidiary beats at different,
faster, beat frequencies between other pairs of harmonics, but because these are
of lower amplitudes, the main and slowest beat just referred to is the one the
ear will latch onto).By adjusting
one of the pipes, this main beat will eventually disappear together with all the
others, and the two are then in tune.

Tuning
Fifths

The
situation gets a further step more complicated when tuning two pipes which are
an interval of a fifth apart.It is
necessary to do this to “lay the bearings” or “set the temperament” when
tuning an organ.How can the
necessary slow, noticeable beats arise in this case, when the fundamental
frequencies of the two pipes are not the same (as they were in the ‘unison’
case), or not exactly an octave apart (as they were in the ‘octave’ case)?

The
main beat in this case, which becomes more and more noticeable (and slower and
slower) as the pipes are pulled progressively more closely into tune, arises
between the third harmonic of the lower note and the second harmonic of the
upper note.Take the example of the fifth formed by playing the G below
middle C and the D above middle C.For
an instrument tuned to the usual modern standard where the A above middle C has
a fundamental frequency of 440.00 Hz, and assuming Equal Temperament, then the
fundamental frequency of the G is 196.00 Hz and that of the D is 293.66 Hz.

The
third harmonic of the lower note, G, is therefore at 196.00 x 3 = 588.00 Hz.The second harmonic of the upper note, D, is at 293.66 x 2 = 587.32 Hz.See how close these two frequencies are!That is why the ear is easily able to detect the beating between them.The frequency difference is minute, 0.68 Hz, the same as in the sine wave
example we opened the discussion with.That
is why I chose these two apparently peculiar numbers for that example.All the fifths in equal temperament have a slowish beat (none are tuned
pure), although the actual beat rate for this G-D interval varies depending on
which octave you are playing it in.If
this fifth was to be tuned pure, you would tune it until no beat could be heard
at all, and in that case there would then be a multiplier of exactly 1.5 between
the two fundamental frequencies – in other words, they would have a frequency
ratio of exactly 3:2.Hence the
love of the ancient Greeks for pure intervals, which are represented by ratios
of exact integers.They discovered
this by observing the lengths of pipes and strings tuned to these intervals.

The
reason why the frequency of the sine waves in the earlier example corresponded
to treble D, an octave above the upper note of the fifth (middle D) we are now
discussing, is because it is the frequency of the second harmonic of that upper
note.This is the same as the
fundamental frequency of the note an octave above the upper note, i.e. treble D.

We
can now see why Lehman’s explanation [3] for how the interval of a fifth
creates a beat was incorrect.He said that “upper harmonics of both notes have a
frequency that is almost identical, at some point several octaves above the
fundamental frequency”.We have
shown that it is not “several octaves” above the fundamental at all, but
only one octave above the fundamental in the case of the upper note, and one and
a half octaves in the case of the lower.

Acoustic
Bass

Unlike
the octave, the interval of a fifth also generates a beat between the
fundamental frequencies as well as between the upper harmonics.This is the case even when the interval is exactly tuned (pure).The reason lies in the arithmetic, because the difference between the two
fundamental frequencies of a fifth equals half the frequency of the lower note,
rather than coinciding with one of those frequencies as in the case of the
octave.However the beat is far too
fast to be perceived except when the two fundamental frequencies are very low,
as in the case of fifths played on 16 foot pedal stops in the lowest octave.Then the beat frequency of the fifth is the same as that of a 32 foot
stop, hence the use of this “quinting” technique to derive an acoustic bass.

However
this brings us onto yet another aspect of beats which creates much confusion.There is absolutely no acoustic energy in the air in any beat regardless
of its frequency, including at the frequencies generated by a quinted acoustic
bass, and that is the main reason why an acoustic bass is so unsatisfactory.It is also the reason why you cannot usually hear a beat as a separate sound at
any frequencies other than very low beat frequencies, when the ear merely
becomes able to follow the time pattern of the envelope of the sound as in the
picture above.Energy would only appear at the beat frequencies if the
propagation of sound in the atmosphere was nonlinear, which it is not.In those circumstances the mechanism would become one of amplitude
modulation of one frequency by another rather than beat formation.In amplitude modulation true frequency sidebands are generated whose
energy has been robbed from the generating waves.The two processes are very different.

Even
so, but rarely, you can hear the actual beat frequencies, and this occurs
when the generating sounds are very loud. Under these conditions the small
nonlinearities in the ear itself cause energy to appear at the beat frequencies
within our hearing mechanism, both the sum and difference frequencies, and you
can hear them as faint tones even though they do not actually exist in the sound
wave impinging on your ear. It is interesting that it is musicians and
musical instrument technicians who usually speak as though these spurious
frequencies are more common than they actually are. If you play an
instrument for your living, you live close to the sound generating mechanism
much of the time, and its sheer loudness is such that you will often hear the
sum and difference frequencies. The same applies to an organ tuner or
voicer, who hears the beats as separate tonal entities between, say, the ranks
of a mixture when close to the pipes. I think this factor is responsible
for such people in the trade assuming that their audiences will always hear them
also, though generally they do not because they are much further away from the
sound source and therefore the sum and difference tones vanish into
inaudibility. This also means that oft-quoted statements such as
"many difference and addition tones is the thing that makes a fine organ
chorus more than equal to the sum of its parts" [13]
apply more to the tuner, voicer and possibly the player than to Joe Public away
down in the body of the church.

Can
you tune using beats when there are no upper harmonics?

No.You could not tune two pure tones – two sine waves – using beats
except when tuning them to the same frequency.Trying to tune them in octaves or fifths is impossible using beats
because the beats are not formed.There
are no harmonics to form them with.Fortunately,
pure sine waves do not arise in nature as they can only be generated
electrically.

However,
the converse problem can also occur if there are too many harmonics.Reed stops have large numbers of harmonics whose strengths diminish only
slowly against harmonic number.If
you try to tune two reed stops in octaves and fifths, there is a cacophony of
strong multiple beats between all these harmonics which can confuse the ear.This is why organs are first tuned using a diapason type of stop, whose
harmonics are well suited both in number and relative strength for beat
formation.Having got the flue work
in tune, the reeds are then tuned in unison with the flues, note by note and
stop by stop.

The
End - (well, the end of this Appendix)

At
the end of this rather exhausting and surprisingly long Appendix on beats, do
you now agree that the average organ builder, organist or composer in Bach’s
time would have been unlikely to understand all about them?Admittedly, there is nothing particularly difficult provided we keep a
clear head and (more importantly) grant that terms such as sine waves,
harmonics, spectra, difference frequencies and nonlinear processes were known
and widely understood.But that
would not and could not have been the case.They would have known about tuning by beats at a practical level in those
days, but I suspect the reasons why beats occurred would have seemed rather
mysterious to most of those who thought about it at all.Therefore, for modern Temperamentalists to imagine that they can
transplant today’s level of understanding backwards two or three centuries can
only be a nonsense.

The
degree of precision required in numbers and arithmetic operations to do with
temperament arises as follows.

Consider
the process of tuning by beats.Let
there be two flue pipes, one of which is already in tune and one which is to be
tuned to it.Tuning becomes progressively more critical and difficult the
higher the frequency because the beats, which are frequency differences, become
faster as those frequencies increase for a given change in length of the pipes.

A
verdict of “pretty well in tune” would probably be given if there was, say,
one beat in around ten seconds for any pair of pipes.Although stricter criteria could be adopted there is no point in making
things too difficult, partly because of the tuning drift which occurs naturally
due to temperature variations etc after an organ has been carefully tuned.Therefore, using this criterion, the two pipes would have to be tuned
until their fundamental frequencies did not differ by more than about 0.1 Hz,
because one beat in ten seconds implies a frequency difference of 0.1 Hz.

Remembering
that tuning is most critical in the upper reaches of the compass, consider the
fifth C on the keyboard, i.e. the C below top C on a 61-note organ keyboard.Even higher notes could be chosen, but again we have to adopt reasonable
parameters if the discussion is to remain sensible and practical.The fundamental frequency of this note on an 8 foot stop is 1046.5 Hz for
an organ tuned to A = 440.00 Hz in Equal Temperament.For simplicity we shall use the approximate figure of 1000 Hz.

A
frequency tolerance of about 0.1 Hz at a frequency of about 1000 Hz implies a
tuning accuracy of the order of 0.0001 or 0.01%.This is therefore also the precision required in temperament calculations
which have to deliver the frequencies of the notes in a particular temperament.
But because there are usually several steps in the calculation of each
frequency, it is necessary that the numerical precision of the numbers used in
each step is greater than that required in the final answer, otherwise the
answer will not be accurate enough owing to truncation or rounding errors.Therefore at least one more significant figure is required throughout the
calculations, meaning that numbers must be represented to at least a precision
of 0.00001 or 0.001%.This is the
same as a precision of 1 part in 100,000, or 6 significant figures, as stated in
the main body of this article.

The
example below shows how the two numbers 531441 (the numerator) and 524288 (the
denominator) can be divided using the age-old method of long division to give
the answer 1.01364.This is the
minimum precision, 6 significant digits, required to represent numbers when
doing temperament arithmetic (Appendix 2).Desirably it would be better to work to at least one further digit so
that the 6 digit answer could be rounded rather than truncated.No problem with an electronic calculator, though I am sure you will agree
that the enthusiasm to do this by hand would rapidly wane.

The
method works by first subtracting the denominator repeatedly from the numerator
until no further subtractions can be done.The denominator is then divided by 10 and subtracted repeatedly from the
remainder left after the first stage.The
whole process is then repeated, dividing the denominator by a further 10 each
time, until the desired degree of precision has been reached.(A computer or calculator still does exactly the same when you ask it to
divide two numbers, except that the numbers are represented in binary form
rather than decimal, therefore the intermediate divisions are by two rather than
ten).

Bach
and his unfortunate contemporaries would almost certainly have been taught how
to do this because there was no alternative method at the time.Even so, I doubt it would have developed into one of their favourite
pastimes and therefore I contend that studies of temperament would have been
largely qualitative in those days rather than quantitative as today, with
serendipity playing a more dominant role than many current Temperamentalists
grant.Even for those with the
determination, the opportunities for error would have been so great that, again,
fully quantitative numerical studies of temperament would have been the
exception rather than the rule.

The
method was still being taught and examined when I was at junior school in the
1950’s.A few years later I was
taught the use of logarithms to do the same job, although the business of
looking up the numbers in log tables, doing the necessary arithmetic on them,
and then reconverting the answer using antilog tables was so involved that I
doubt it was really very much easier.There
was also still far too much opportunity for error.The arrival of electronic hand held calculators in the 1970’s finally
swept away all this centuries-old arithmetical baggage the human race had
carried for so long.

2.
“Here’s to Plato”, Rectorial Address to the University of Dundee, Stephen
Fry, 1 November 1995.

3.
“Bach’s extraordinary temperament : our Rosetta Stone”, Bradley Lehman, Early
Music, February 2005. (Lehman, and presumably his publisher, were apparently
unaware of previous material by Sparschuh and Zapf which had covered much of the
same ground. For more on this, see subsequent correspondence such as the
letter by Mobbs and Mackenzie of Ord, Early Music, August 2005).

11.
Some might object to intervals such as A# to F being called a fifth.
However, because of the equivalence of the notes played in the cycle of twelve
fifths to the twelve semitones of a chromatic scale with any starting note, no
concept of key was implied in this discussion and none should be assumed.
It is only necessary here to identify the physical notes on the keyboard in some
convenient manner, and
the tuner's convention of referring to the white notes as naturals and the black
ones as sharps seems as good as any. The point is simply that a fifth
consists of seven adjacent semitones (eight if counting the starting note). Using MIDI note numbers would have
served just as well, but it would have caused unnecessary difficulties for the
uninitiated.

12.
With his laws of motion and his calculus, Newton proved in the mid-17th century that the time period of a simple plumb-line type of
pendulum is proportional to the square root of its length. This statement is true provided the mass of the string is
negligible compared to that of the bob it supports, that the dimensions of
the bob are negligible compared to the length of the string, and that the arc
of swing is small (typically ten degrees or so). In these circumstances a "seconds pendulum",
which takes one second to swing from one extremity to the other, is almost
exactly one metre long, hence the size of long case clocks which appeared
shortly afterwards. Such a pendulum could easily have been carried
around for tuning and other musical purposes. It is difficult to see how
else beat rates could have been measured accurately at the keyboard at the time we are
considering in this article, until experience had maybe conferred a sufficiently
accurate mental notion of time on an experienced tuner.

13.
In "The Mixture - to be taken as before?", S Bicknell, number 3 of 6
articles published in 'Choir and Organ' in 1998-9 under the heading 'Spit and
Polish'.

Probably
stimulated by proposals put forward earlier by Galileo, Mersenne made the first
documented, but rather crude, determination of the absolute frequency of a
musical note. This was at 84 Hz, which lies between bottom E and F on an 8 foot
stop of a modern organ tuned to A = 440 Hz in ET. He first timed the
motion of a long heavy wire that moved slowly enough to be followed by eye. From
theoretical considerations based on his own laws describing the pitch of a taut
string, he could then calculate - but not measure directly - the approximate
frequency of a shorter and lighter wire, one that produced an audible
sound. Note that this was a rather shaky application of logic in that his
"laws" themselves remained hypothetical until such time as any
frequency could be measured to verify them. However, even neglecting this
problem, the result was a long way from the accuracy required for actual studies
in temperament as we pursue them today, and therefore it does not conflict with
my conjecture that the concept of frequency and the part played by it in musical
acoustics was not widespread in the 17th and 18th centuries. How could it
have been, when the foregoing shows it was so difficult to measure and when
education - the dissemination of such knowledge - was so limited both in terms
of availability and content?

15.
J-P Bacconet, private communication, December 2006.

16.
"Keyboard Temperaments with Impure Octaves", C E Pykett, 2008.
Currently available on this website (read).